Linear Momentum Formula: Dimensional Formula, Conservation

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Linear momentum is defined as the product of a mass of an object to its velocity. It is a vector quantity. It is represented by the symbol p. The direction of the momentum of a body is the same as the direction of its velocity. Momentum is a conserved quantity, which is to say the total momentum of an isolated system remains constant. The SI unit of linear momentum is kg m/s.

The formula for the linear momentum of a body is given by

p = m⋅v

Where,

m = mass of the object

v = velocity of the object

Table of Content

Dimensional Formula of Linear Momentum
Conservation of Linear Momentum
Things to Remember
Sample Questions

Key Takeaways: Linear Momentum, Dimensional Formula, Conservation of Linear Momentum, Centre of Mass

Dimensional Formula of Linear Momentum

Now, linear momentum is calculated using the formula,

linear momentum = mass × velocity

So, the dimensional formula of linear momentum can be calculated using the above formula.

Linear Momentum Formula

Linear Momentum Formula

Dimensional formula of mass = [M¹L⁰T⁰]

Dimensional formula of velocity = [M⁰L¹T^-1]

Dimensional Formula of linear momentum = [M¹L⁰T⁰] × [M⁰L¹T^-1] = [M¹L¹T^-1]

Therefore, the dimensional formula of linear momentum is [M¹L¹T^-1].

Also Read: What is Unit of Momentum? - SI & Other Units

Conservation of Linear Momentum

The conservation of linear momentum of an object can be explained using Newton’s second law of motion. And according to this law, the rate of change of linear momentum of an object is equal to the net force acting on that object.

F = dpdt

We know, p = m⋅v

Differentiating this equation on both sides, we get

dpdt= mdvdt= ma = F_ext

Ifdpdt= 0, then p is constant.

Now, if the net external force on a system is zero, then the linear momentum is conserved and the velocity of the centre of mass remains constant.

Conservation of Momentum: Example

Conservation of Momentum: Example

Things To Remember

Linear momentum is a vector quantity that is calculated as the product of a mass of an object to its velocity.
The SI unit of linear momentum is kg m/s.
The dimensional formula of linear momentum is [M¹L¹T^-1].
The direction of linear momentum is in the direction of the velocity of the object.
The linear momentum of a system remains constant if the net external force acting on it is zero. This means that the momentum is conserved.

Also Read:

Angular Momentum	Angular Momentum of Electron	Coefficient of Linear Expansion

Sample Questions

Ques: Determine the linear momentum of a body with a mass of 5 kg and moving with a velocity of 50 m/s. (2 marks)

Ans: m = 5 kg

v = 50 m/s

Linear momentum, p = m⋅v

p = 5 × 50 = 250 kg m/s

The linear momentum of this body is 250 kg m/s.

Ques: If the linear momentum of an object moving with a velocity of 25 m/s is 100 kg m/s, then find out the mass of the object. (3 marks)

Ans: v = 25 m/s

p = 100 kg m/s

The formula for linear momentum is p = m⋅v

100 = m × 25

m = 4 kg

So, the mass of this object is 4 kg.

Ques: Explain the conservation of linear momentum. (2 marks)

Ans: The law of conservation of linear momentum states that if the net external force on a system is zero, then the linear momentum is conserved. The linear momentum of the individual particles in the system keeps changing, but the overall linear momentum of the system remains constant unless an external force acts on it.

Ques: An object of mass 10 kg is at rest. Calculate its linear momentum. (2 marks)

Ans: Here, the given object of mass 10 kg is at rest, which means its velocity is zero.

Now, we know linear momentum is calculated as p = m⋅v

p = 10 × 0 = 0

Thus, the linear momentum of the given object is zero.

Ques: A child sits motionless at the end of a long trolley moving on a smooth horizontal path with uniform speed. If the child runs about the trolley randomly, then what is the speed of the centre of the mass of the system? (2 marks)

Ans: When the child starts running about the trolley in any manner, the speed of the centre of mass of the system remains unchanged. Here, the child and the trolley serve as a single system and the forces applicable are only internal. So, the momentum of the system is conserved and hence, the velocity remains the same.

Ques: Is the total momentum of the universe constant? Explain. (2 marks)

Ans: The total momentum of the universe remains constant. There is no system larger than the universe comprising it. So, there is no external force acting on the universe and hence, its linear momentum is conserved.

Ques: If the linear momentum of an object with a mass of 2 kg is 36 kg m/s, then calculate its velocity. (3 marks)

Ans: m = 2 kg

p = 36 kg m/s

Linear momentum, p = m⋅v

36 = 2 × v

v = 18 m/s

So, the velocity of this object is 18 m/s

CBSE CLASS XII Related Questions

1.
(I) The real image of an object placed between $f$ and $2f$ from a convex lens can be seen on a screen placed at the image location. If the screen is removed, is the image still there? Explain.
(II) Plane and convex mirrors produce virtual images of objects. Can they produce real images under some circumstances? Explain.

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2.
A rectangular loop of size 5 cm $\times$ 8 cm is lying in the $x$-$y$ plane in a uniform magnetic field $\vec{B} = (2.0\ \text{T})\hat{k}$. The total magnetic flux linked with the loop is:

80 Wb
16 Wb
$8 \times 10^{-2}$ Wb
$8 \times 10^{-3}$ Wb

View Solution

3.
A parallel beam of monochromatic light falls normally on a single slit of width $a$ and a diffraction pattern is observed on a screen placed at distance $D$ from the slit. Explain:
(I) the formation of maxima and minima in the diffraction pattern, and
(II) why the maxima go on becoming weaker and weaker with its increasing number $n$.

View Solution

4.
Consider a cylindrical conductor of length $l$ and area of cross-section $A$. Current $I$ is maintained in the conductor and electrons drift with velocity $\vec{v}_d$ ($|\vec{v}_d| = \frac{e|\vec{E}|}{m}\tau$), (where symbols have their usual meanings). Show that the conductivity $\sigma$ of the material of the conductor is given by
$$\sigma = \frac{ne^2}{m}\tau.$$

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5.
In a region, the electric potential varies as $V = 10 - 50x$, where $V$ is in volts and $x$ is in meters. The electric field in the region is

$10\ \text{N/C}$ along $+x$
$10\ \text{N/C}$ along $-x$
$50\ \text{N/C}$ along $+x$
$50\ \text{N/C}$ along $-x$

View Solution

6.
The resistance of a metal wire at $20^\circ\text{C}$ is $1.05\,\Omega$ and at $100^\circ\text{C}$ is $1.38\,\Omega$. Determine the temperature coefficient of resistivity of this metal.

View Solution

Linear Momentum Formula: Dimensional Formula, Conservation

Dimensional Formula of Linear Momentum

Conservation of Linear Momentum

Things To Remember

Sample Questions

CBSE CLASS XII Related Questions

2.
A rectangular loop of size 5 cm $\times$ 8 cm is lying in the $x$-$y$ plane in a uniform magnetic field $\vec{B} = (2.0\ \text{T})\hat{k}$. The total magnetic flux linked with the loop is:

5.
In a region, the electric potential varies as \(V = 10 - 50x\), where \(V\) is in volts and \(x\) is in meters. The electric field in the region is

6.
The resistance of a metal wire at \(20^\circ\text{C}\) is \(1.05\,\Omega\) and at \(100^\circ\text{C}\) is \(1.38\,\Omega\). Determine the temperature coefficient of resistivity of this metal.

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Linear Momentum Formula: Dimensional Formula, Conservation

Dimensional Formula of Linear Momentum

Conservation of Linear Momentum

Things To Remember

Sample Questions

CBSE CLASS XII Related Questions

2.A rectangular loop of size 5 cm $\times$ 8 cm is lying in the $x$-$y$ plane in a uniform magnetic field $\vec{B} = (2.0\ \text{T})\hat{k}$. The total magnetic flux linked with the loop is:

5.In a region, the electric potential varies as \(V = 10 - 50x\), where \(V\) is in volts and \(x\) is in meters. The electric field in the region is

6.The resistance of a metal wire at \(20^\circ\text{C}\) is \(1.05\,\Omega\) and at \(100^\circ\text{C}\) is \(1.38\,\Omega\). Determine the temperature coefficient of resistivity of this metal.

Comments

SUBSCRIBE TO OUR NEWS LETTER

2.
A rectangular loop of size 5 cm $\times$ 8 cm is lying in the $x$-$y$ plane in a uniform magnetic field $\vec{B} = (2.0\ \text{T})\hat{k}$. The total magnetic flux linked with the loop is:

5.
In a region, the electric potential varies as \(V = 10 - 50x\), where \(V\) is in volts and \(x\) is in meters. The electric field in the region is

6.
The resistance of a metal wire at \(20^\circ\text{C}\) is \(1.05\,\Omega\) and at \(100^\circ\text{C}\) is \(1.38\,\Omega\). Determine the temperature coefficient of resistivity of this metal.